Existence Of Positive Solutions For Boundary Value Problem Of Three-order Delay Differential Equations

Boundary value problem is always one of the significant problems in the research field to differential equations. Many research works have been done in this direction(see for instance[1, 2, 7, 12, 13, 14, 21, 22, 23, 28]. But by comparison, the research to boundary value problems to delay differential equations is relatively less.In recent years, due to the delay differential equations, applications in various field is increasingly widespread, boundary value problem to delay differential equations has aroused the concern of many mathematicians . Now a series of important results have achieved in this direction (see for instance[3, 10, 16, 24, 26, 27]). Other results(see for instance [5, 11, 15, 19, 20, 25])In 1994, Erbe[7] considered the positive solutions to the boundary value problemx″(t)+a(t)f(x) = 0, 01) f : [0,∞)→[0,∞)is continuous;(A₂) a : [0,1]→[0,∞)is continuous and does not vanish identically on any subinterval;In 1999, Jiang [16] studied the positive solutions to the following boundary value problemx″(t) + g(t, x(t-Ï„))=0, 0 < t < 1,Ï„> 0, x(t) = 0,-τ≤t≤0, x(1) = 0,They assumed that 0 <Ï„< 1/4 and g(t, 0) > 0.In 2005, Bai and Xu [3] studied the following boundary value problemx″(t) +λg(t, x(t-Ï„)) = 0, 0 < t < 1,Ï„> 0 , x(t) =0, -τ≤t≤0, x(1) = 0,they assumed that(B₁) 0 <Ï„< 1/2 ;(B₂) g(t,x) = a(t)f(t,x), a : (0,1)→[0,∞)is continuous , and f : [0,1]×[0,∞)→[0,∞)is conuntinupus;(B₃)∫₀¹s(1-s)a(s)dx<∞, (?)θ∈[Ï„, 1/2), such that∫_θ+Ï„^1-θ+Ï„ a(s)ds > 0. In 2000, Ma[22] studied the following boundary value problemx″(t) +λh(t)f(x) = 0, 0 < t < 1, x(0) = 0, ax(η) = x(1),where(C₁)λis a positive parameter;η∈(0, 1)and 0 < aη< 1;(C₂) a : [0,1]→[0,∞) is continuous and does not vanish identically on any subset of positive measure;(C₃) f:[0,∞)→[0,∞) is continuous, f_∞=(?) f(x)/x =∞.In 2007, Wang and Shen [24] studied the boundary value problemx″(t) +λa(t)f(t, x(t -Ï„)) = 0, 0≤t≤1, x(t) =0, -τ≤t≤0, x(1) = ax(η),they assumed that(D₁) 0 <η< 1, 0 < aη< 1, 0 <Ï„< 1;(D₂) p : (0,1)→[0,∞)is continuous,and f : [0,1]×[0,∞)→[0,∞)is continuous;(D₃) f : [0,1]×[0,∞)is continuous , 0 <∫₀¹s(1-s)p(s)dt <∞, and (?)0≤bb+Ï„^c+Ï„p(t)dt>0.In 2005, Sun [23] considered the following boundary value problemx(?)(t) -λa(t)f(t,x(t)) = 0, 0 < t < 1, x(0) = x'(η) = x″(1) = 0,and concluded sufficient condition of the existence of positive solutions for the caseη∈[1/2, 1),λis a positive parameter; a(t) : (0,1)→[0,∞)is continuous. f : [0,1]×[0,∞)→[0,∞)is continuous. In this paper,we consider the existence of positive solutions for boundary value problem of three-order delay differential equationx(?)(t)-λa(t)f(t,x(t-Ï„))=0, t∈J(?)[0,1], (1)x(t) = 0, -τ≤t≤0, (2)x′(η) = x″(1) = 0, (3)we assumed that(H₁)λis a positive parameter,η∈[0, 1/2) is a constant;τ∈[0,Ï„₀],Ï„₀ < 1 is a constant;(H₂) f : [0,1]×[0,∞]→[0,∞] is continuous and satisfying the condition Carathéodary;(H₃) a(t) is a nonegative continuous function defined on (0,1), and there exist constants 0b+Ï„^c+Ï„ f(x)dx > 0 . We shall present some sufficient condition withλbelonging to different open interval, prove that the operator have fixed-points by Guo-Krasnoseklkii fixed-point theorem, therefore the boundary value problem existence positive solutions.定理2.1 (Guo-Krasnoselskii)[8] Let E is a real Banach space and K (?) E is a cone in E. AssumeΩ₁ andΩ₂ are open subset of E with 0∈Ω₁ and (?)₁ (?)Ω₂. Let T : (?)₂\Ω₁→E be a continuous operator. In addition suppose either (E₁)x∈(?)Ω₁(?)‖Tx‖≤‖x‖；x∈Ω₂(?)‖Tx‖≥‖x‖；or(E₂)x∈(?)Ω₁(?)‖Tx‖≥‖x‖；x∈(?)Ω₂(?)‖Tx‖≤‖x‖.hold. Then T has a fixed point in (?)₂\Ω₁.定理2.2 (Arzelà- Ascoli)[18]Every bounded and equicontinuous sequence of function F = {f(t)} in [α,β] contains a subsequence that converge uniformly in [α,β]引理2.1 Letθ= min{b, 1-c},0≤b3(0,1), if x(?)(t) > 0,t∈J, then x(t) satisfiesDefine a coneDefine a integral operator T : K→K引理2.2 The fixed-point of T is a solution of the boundary value problem.引理2.3 T : K→K is completely continuous operator. Let定理2.3 Let (H₁) - (H₃) hold and f_∞> 0,f⁰ <∞, then there exists at least one positive solution to the boundary value problem (1)-(3) for定理2.4 Let (H₁) - (H₃) hold and f^∞<∞, f₀ > 0, then there exists at least one positive solution to the boundary value problem, (1)³ for 定理2.5 Let (H₁) - (H₃) hold and f_∞=∞,f₀ =∞,and there exist two positive constantsα,βsuch thatthen the boundary value problem (1)-(3) has at least two positive solutions x₁(t),x₂(t) , with 0 <‖x₁‖<α<‖x₂‖, for any 0 <λ≤β.